Frequency method for determination self oscillations in control systems with a piezo actuator for astrophysical research

doi:10.15406/aaoaj.2024.08.00198

eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Research Article Volume 8 Issue 2

Frequency method for determination self oscillations in control systems with a piezo actuator for astrophysical research

Afonin SM

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National Research University of Electronic Technology, Russia

Correspondence: Afonin SM, National Research University of Electronic Technology MIET, Moscow, Russia

Received: June 20, 2024 | Published: July 2, 2024

Citation: Afonin SM. Frequency method for determination self-oscillations in control systems with a piezo actuator for astrophysical research. Aeron Aero Open Access J. 2024;8(2):115-117. DOI: 10.15406/aaoaj.2024.08.00198

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Abstract

For the control system with a piezo actuator in astrophysical research the condition for the existence of self-oscillations is determined. Frequency method for determination self-oscillations in control systems is applied. By using the harmonious linearization of hysteresis and Nyquist stability criterion the condition of the existence of self-oscillations is obtained.

Keywords: frequency method, control system, piezoactuator, hysteresis, self-oscillations, astrophysical research

Introduction

A piezo actuator is used in astrophysics for image stabilization and scan system.^1–19 Frequency method for determination self-oscillations in scan system is applied.^20–46 for Nyquist stability criterion of self-oscillations at harmonious linearization of hysteresis characteristic of a piezo actuator.

Condition of self-oscillations

The scan system with a piezo actuator is used for astrophysical research in system adaptive optics. Nyquist stability criterion of self-oscillations at harmonious linearization of hysteresis characteristic^2,20–40 of a piezo actuator has the form

$W_{l} (α Ω) W_{g} (E_{m max}) = - 1$ $W_{l} (α Ω) W_{g} (E_{m max}) = - 1$

where α is the imaginary unit, Ω - the frequency of self-oscillations, $W_{l} (α Ω)$ $W_{l} (α Ω)$ - the frequency transfer function of the linear part, $W_{g} (E_{m max})$ $W_{g} (E_{m max})$ - the transfer function of the hysteresis part, $E_{m max}$ $E_{m max}$ - amplitude of the electric field strength for m axis.

For the scan system with a piezo actuator for astrophysical research the condition of self-oscillations is written

$1 + W_{l} (α Ω) W_{g} (E_{m max}) = 0$ $1 + W_{l} (α Ω) W_{g} (E_{m max}) = 0$

The condition of self-oscillations is determined in the form

$W_{l} (α Ω) = - \frac{1}{W_{g} (E_{m max})}$ $W_{l} (α Ω) = - \frac{1}{W_{g} (E_{m max})}$

here the left side of this equation has the form of the amplitude-phase characteristic of the linear part of the system, and the right side of the equation has the form of the inverse amplitude-phase characteristic of the hysteresis link of the piezo actuator with the inverse sign minus.

Preisach hysteresis function a piezo actuator has the form^22-40

$S_{i} = F {[{E_{m} |}_{0}^{t}, t, S_{i} (0), sign {˙ E}_{m}]}$ $S_{i} = F {[{E_{m} |}_{0}^{t}, t, S_{i} (0), sign {\dot{E}}_{m}]}_{}^{}$

here $t, S_{i}, S_{i} (0), E_{m}$ $t, S_{i}, S_{i} (0), E_{m}$ and $sign {˙ E}_{m}$ $sign {\dot{E}}_{m}$ - the time, the deformation, the initial deformation,, the strength of electric field and the sign velocity.

The symmetric hysteresis the deformation^22-40 a piezo actuator has the form

$S_{i} = d_{m i} E_{m} - γ_{m i} E_{m max} {(1 - \frac{E_{m}^{2}}{E_{m max}^{2}})}^{n} sign {˙ E}_{m}$ $S_{i} = d_{m i} E_{m} - γ_{m i} E_{m max} {(1 - \frac{E_{m}^{2}}{E_{m max}^{2}})}^{n} sign {\dot{E}}_{m}$

$d_{m i} = d_{m i}^{0} + a_{m i} E_{m}^{2}, γ_{m i} = S_{i}^{0} / E_{m max}$ $d_{m i} = d_{m i}^{0} + a_{m i} E_{m}^{2}, γ_{m i} = S_{i}^{0} / E_{m max}$

here $d_{m i}, γ_{m i}, S_{i}^{0}, n$ $d_{m i}, γ_{m i}, S_{i}^{0}, n$ , - the piezo module, the hysteresis coefficient, the relative deformation for $E_{m} = 0$ $E_{m} = 0$ , and the power 1, 2, 3, ….

The transfer function of the linear part of the scan system with a piezo actuator for elastic-inertia load^22,37-46 has the form

$W_{l} (p) = \frac{k_{l}}{T_{t}^{2} p^{2} + 2 T_{t} ξ_{t} p + 1}$ $W_{l} (p) = \frac{k_{l}}{T_{t}^{2} p^{2} + 2 T_{t} ξ_{t} p + 1}$

After transformations we have this condition for the scan system with the PZT actuator at the power in the form

$\frac{1}{\frac{1 - T_{t}^{2} Ω^{2}}{k_{l}} + α \frac{2 T_{t} ξ_{t} Ω}{k_{l}}} = \frac{1}{- (d_{m i}^{0} + a_{m i} E_{m max}^{2}) + α \frac{8 γ_{m i}}{3 π}}$ $\frac{1}{\frac{1 - T_{t}^{2} Ω^{2}}{k_{l}} + α \frac{2 T_{t} ξ_{t} Ω}{k_{l}}} = \frac{1}{- (d_{m i}^{0} + a_{m i} E_{m max}^{2}) + α \frac{8 γ_{m i}}{3 π}}$

$Ω = \frac{4 γ_{m i} k_{l}}{3 π T_{t} ξ_{t}}$

For the the scan system with the PZT actuator = 3.2×10⁸ V/m, $d_{33}^{0}$ = 4×10^-10 m/V, $γ_{33}$ = 0.8×10^-10 m/V, $a_{33}$ = 3.1×10^-22 m³/V³, $T_{t}$ = 10^-3 s, $ξ_{t}$ = 10^-2 the frequency is determined $Ω$ =1.1×10³ s^-1 with error of 10 %.

The frequency transfer function of the symmetric hysteresis the deformation of a piezo actuator is received in the form

$W_{g} (E_{m max}) = S_{i} (E_{m max}) / E_{m} (E_{m max})$

$W_{g} (E_{m max}) = q_{m i} (E_{m max}) + α {q^{'}}_{m i} (E_{m max})$

For n = 1

$q_{m i} (E_{m max}) = d_{m i}, {q^{'}}_{m i} (E_{m max}) = - \frac{4 \cdot 2 \cdot γ_{m i}}{π \cdot 3} = - \frac{8 γ_{m i}}{3 π}$

For n = 2

$q_{m i} (E_{m max}) = d_{m i}, {q^{'}}_{m i} (E_{m max}) = - \frac{4 \cdot 2 \cdot 4 \cdot γ_{m i}}{π \cdot 3 \cdot 5} = - \frac{32 γ_{m i}}{15 π}$

For n = 2

$q_{m i} (E_{m max}) = d_{m i}, {q^{'}}_{m i} (E_{m max}) = - \frac{4 \cdot 2 \cdot 4 \cdot 6 \cdot γ_{m i}}{π \cdot 3 \cdot 5 \cdot 7} = - \frac{192 γ_{m i}}{105 π}$

For n to n + 1

$q_{m i} (E_{m max}) = d_{m i}, {q^{'}}_{m i (n)} (E_{m max}) = \frac{2 n}{2 n + 1} {q^{'}}_{m i (n - 1)} (E_{m max})$

For n + 1

$q_{m i} (E_{m max}) = d_{m i}, {q^{'}}_{m i} (E_{m max}) = - \frac{4 \cdot 2 \cdot 4 \cdot 6 \cdot ... \cdot 2 n \cdot γ_{m i}}{π \cdot 3 \cdot 5 \cdot 7 \cdot ... \cdot (2 n + 1)}$

The stability criterion and frequency method are used.

Discussion

By using of frequency method the parameters of self-oscillations are obtained in the scan system. Nyquist stability criterion is used for calculation the self-oscillations in the control system with a piezo actuator at harmonious linearization of hysteresis characteristic of a piezo actuator.

Conclusion

For the scan system its condition of self-oscillations is determined. For calculation the self-oscillations frequency method is applied at harmonious linearization of hysteresis characteristic of a piezo actuator.